1

Improving Worst-Case End-to-End Delay Analysisof Multiple Classes of AVB Traffic in TSN

Networks using Network CalculusLuxi Zhao, Paul Pop, Zhong Zheng, Hugo Daigmorte and Marc Boyer

Abstract—Time-Sensitive Networking (TSN) is a set of amend-ments that extend Ethernet to support distributed safety-criticaland real-time applications in the industrial automation, aerospaceand automotive areas. TSN integrates multiple traffic types, inwhich one solution is supporting Time-Triggered (TT) trafficscheduled based on Gate-Control-Lists (GCLs), Audio-Video-Bridging (AVB) traffic according to IEEE 802.1BA that hasbounded latencies, and Best-Effort (BE) traffic, for which noguarantees are provided. This paper proposes an improved timinganalysis method reducing the pessimism for the worst-case end-to-end delays of AVB traffic by considering the limitations fromthe physical link rate and the output of Credit-Based Shaper(CBS). Moreover, the paper extends the timing analysis methodto multiple AVB classes and proofs the AVB credit bounds of CBSfor multiple classes, which are prerequisites for non-overflow ofCBS credits and preventing starvation of AVB traffic. Finally,we evaluate the improved analysis method on both syntheticand real-world test cases, showing the significant reduction ofpessimism on latency bounds compared to related work, andthe correctness validation compared with simulation results.Additionally, we evaluate the scalability of our implementationwith variation of the load of TT flows and the number of AVBclasses.

I. INTRODUCTION

ETHERNET is a well-established network protocol thathas excellent bandwidth, scalability, compatibility and

cost properties [1]. However, it is not suitable for real-timeand safety critical applications [2]. Distributed safety-criticalapplications, like those found in the aerospace, automotive andindustrial automation domains, require certification evidencefor the correct real-time behavior of critical communication.Therefore, several extensions to the Ethernet protocol havebeen proposed, such as, ARINC 664 Specification Part 7 [3],TTEthernet [4], and EtherCAT [5]. In 2012, the IEEE 802.1Time-Sensitive Networking (TSN) Task Group [6] has beenfounded to define standard real-time and safety-critical en-hancements for Ethernet.

In this paper we consider a TSN solution supporting Credit-Based Shaper (CBS) (previously defined for Audio-Video

L.X. Zhao and P. Pop are with the Department of Applied Mathematics andComputer Science, Technical University of Denmark, Copenhagen, Denmark,e-mail: luxzha@dtu.dk; paupo@dtu.dk.

Z. Zheng is with the Department of Electronics and Information Engineer-ing, Beihang University, Beijing, China, e-mail: zhengzhong@buaa.edu.cn

H. Daigmorte and M. Boyer are with the Department on ”Infor-mation Processing and Systems” at ONERA, Toulouse, France, e-mail:Hugo.Daigmorte@onera.fr; Marc.Boyer@onera.fr.

Bridging (AVB) in 802.1BA [7], currently in [6, § 8.6.8.2] withthe 802.1Qbv [8] enhancements for Scheduled Traffic (nowin [6, § 8.6.8.4]) that add a Gate-Control Lists (GCLs). Weconsider three traffic-types with varying the criticality: Time-Triggered (TT) traffic (high priority), AVB traffic (mediumpriority) and Best-Effort (BE) traffic (low priority). TT trafficsupports hard real-time applications that require very lowlatency and jitter. It has the highest priority and is transmittedbased on schedule tables called GCLs that rely on a globalsynchronized clock (802.1ASrev [9]). AVB traffic is intendedfor applications that require bounded end-to-end latencies, buthas a lower priority than TT traffic. It uses the CBS to preventthe starvation of lower priority AVB traffic. The TSN TaskGroup has extended AVB architecture, allowing the definitionof multiple CBS, so we consider any number of AVB classes.BE traffic is used for applications that do not require anytiming guarantees and is mapped to the remaining low priorityqueues. We call this architecture TSN/GCL+CBS.

The schedulability of the scheduled TT traffic can be guar-anteed during design phase, by synthesizing the GCLs [10].However, an AVB flow is schedulable only if its worst-caseend-to-end delay (WCD) is smaller than its deadline. More-over, AVB in TSN is envisioned for time-critical applicationsthat require bounded latency, hence it is very important tohave a safe and guaranteed WCD analysis for AVB thattakes into account the TT flows. Although latency analysismethods have been successfully applied to AVB traffic in AVBnetworks [11], [12], [13], [14], [15], they do not take accountfor the interference of TT traffic on the AVB traffic and onlyconsider two or three AVB classes. A Network Calculus-basedanalyses to compute the WCDs of Rate-Constrained (RC)traffic with the consideration of the static scheduled TT framesin TTEthernet have been proposed [16], [17], [18], but thetechniques are not applicable for TSN: RC traffic has no CBS,and TSN schedules TT traffic differently from TTEthernet:individual TT frames are considered in TTEthernet, whereasTSN schedules TT windows may including several TT frames.

The timing analysis of non-TT traffic classes in TSN hasbeen addressed in [19], considering closed-gate blocking, strictpriorities and a “Peristaltic Traffic Shaper”. However, theiranalysis is not applicable to AVB, which uses CBS. [20] givesthe latency bounds for AVB traffic affected by control-data-traffic (CDT) in TSN, but assuming CDT as leaky bucket(LB) or length rate quotient (LRQ), which does not fit for

December 20, 2018

2

Fig. 1. TSN network topology example

the TT traffic. Researchers have extended the Eligible Intervalanalysis [14] to calculate the delay of AVB traffic in TSN [21].However, it only focuses on the latency bounds through asingle node and does not consider relative offsets between TTwindows, but just assumes the TT windows always arrangingback-to-back. The AVB Latency Math equation has beenextended to consider the TT traffic in TSN [22]. However,it does not consider the actual situations for AVB flows in thenetwork, but just assumes that maximum allocable bandwidthis occupied by the corresponding AVB traffic class, thuscausing overly pessimistic, i.e., leading to overly large WCDs.In addition, [22] can only be used to determine the WCDs ofAVB Class A traffic. The initial idea on the AVB analysis inTSN network based on the Network Calculus has been givenby [23], but the WCDs results are pessimistic. Moreover, onlytwo AVB traffic classes are in consideration in [23], but thelatest standard 802.1Q-2018 [24] supports any number of AVBclasses, and adding new classes requires additional proof forassociated credit bounds(1).

The main contributions of our paper are as follows:• We propose a Network Calculus-based method from [23] forthe general case with arbitrary number of AVB traffic classesin a TSN network, which requires a proof of credit boundsof CBS for multiple AVB classes.• We reduce the pessimism of the analysis and this providestighter latency bounds for AVB, by introducing the limitationsfrom the physical link rate and the output of CBS, whichare denoted as the link and CBS shaping curves with theconsideration of TT traffic.• We evaluate the proposed approach on both synthetic andreal-world test cases, and compare it with related workand simulation results to show the significant reduction ofpessimism on latency bounds, and to validate the correctnessand the scalability of our implementation.

The paper is organized as follows. Sect. II presents thearchitecture and application models. Sect. III introduces TSN.Sect. IV briefly introduces the Network Calculus conceptsneeded for the analysis. Sect. V gives the proof of creditbounds for multiple classes of AVB traffic and presents tighterAVB latency bounds by introducing shaping curves. Sect. VIevaluates the proposed analysis and Sect. VII concludes thepaper.

II. SYSTEM MODEL

A TSN network is composed of a set of end systems (ES)and switches (SW) also called nodes, connected via physicallinks. In this paper, we assume, without loss of generality,that all physical links have the same rate C. The links are

(1)Even going from 2 to 3 requires specific work [15].

Fig. 2. TSN/GCL+CBS architecture for an output port in an ES/SW

full duplex, allowing thus communication in both directions,and the networks can be multi-hop. The output port of a SWis connected to one ES or an input port of another SW. Anexample is presented in Fig. 1, where we have 4 ESes, ES1 toES4, and 3 SWs, SW1 to SW3. A dataflow routing dri ∈ R isstatically defined as a directed ordered sequence of physicallinks connecting a single source ES to one or more destinationESes. For example, in Fig. 1, dr1 connects the source endsystem ES1 to the destination end systems ES3 and ES4, whiledr2 connects ES2 to ES3.

The messages running in the ESes communicate via flows,which have a single source and may have multiple destinations.Each source ES is able to send multiple flows to the network.As mentioned, TSN supports three traffic classes: TT, AVBand BE. We assume that the traffic class for each applicationhas been decided by the designer and we define the sets τT T ,τMi , τBE with τ = τT T ∪τMi ∪τBE as the set of all the flows inthe TSN network. The subscripts Mi (i∈ [1,7]) for AVB denoterespectively the different AVB traffic classes. For TT traffic,we know the GCL for TT traffic in each output port h of nodes,i.e., the opening and closing time of TT traffic (denoted withTT window) and GCL period (ph

GCL). For an AVB flow τMi[k] ∈τMi , we know its frame size lMi[k], the minimum frame intervalpMi[k] in the source ES and the traffic class Mi it belongs to.The AVB Class Mi has higher priority than the AVB ClassMi+1. The flows assigned the same priority belong to the sameAVB traffic class Mi, and frames within each traffic class arein FIFO order. Moreover, we know the maximum frame sizelmaxBE of BE traffic.

III. TSN/GCL+CBS OUTPUT PORT

The TSN Task group has defined a lot of mechanisms,which can interact in several ways. In this paper, we assume aspecific configuration, calles TSN/GCL+CBS architecture andwill present how TT and AVB flows are transmitted in thiscase. Fig. 2 gives an illustration for an output port of a node.Each one has eight queues for storing frames that wait to beforwarded on the corresponding link, one or more (Nh

T T ) forTT queues, two or more (Nh

CBS) for AVB queues (respectivelyfor Class Mi) and the remaining queues are used for BE. Everyqueue has a gate with two states, open and closed. Frameswaiting in the queue are eligible to be forwarded only if theassociated gate is open.

The gates for each queue are controlled by a GCL, whichis created offline and contains the times when the associated

3

Fig. 3. Non-preemption integration modes

Fig. 4. CBS example with non-preemption mode

gates are open and closed [10], [25]. It is created such thatwhen associated gate for TT traffic is open, the remaininggates for other traffic (AVB and BE) are closed, and vice versa(aka. exclusive gatig). Therefore, AVB traffic is prevented fromtransmitting in the time windows reserved for TT frames.

In addition, two integration modes are introduced to solvethe issue when an AVB frame is already in transmission at thebeginning of time window for TT, i.e., non-preemption andpreemption modes. In this paper, we only focus on the discus-sion of the non-preemption integration mode (see Fig. 3) andpreemption can be easily extended. The non-preemption modeuses a “guard band” before each TT window to prevent theAVB frame from initiating transmission if there is insufficienttime available to transmit the entirely of that frame before theTT gate open. The non-preemption integration mode will leadto wasted bandwidth due to the guard band, but it ensures nodelays for TT traffic.

An enqueued AVB frame is transmitted only if the associ-ated gate is open and the Credit-Based Shaper (CBS) permits.Each AVB class has an associated credit value and is initializedto zero. When the associated AVB gate is open, the variationof credit is the same as the one in AVB network [7], i.e., de-creasing with a send slope during the transmission of an AVBframe and increasing with an idle slope when AVB framesare waiting to be transmitted due to other higher priority AVBframes or the negative credit. When the AVB gate is closed,the associated credit is “frozen”. This is illustrated for examplein Fig. 4, where we have three classes of AVB traffic Mi(i= 1,2,3), and show the variation of credit for respective AVBclass. Particularly, during guard bands, we assume that thecredit for AVB traffic is also “frozen”, otherwise if increasedmay leading to a credit overflow problem thus causing thestarvation of lower priority AVB traffic [26].

IV. NETWORK CALCULUS BACKGROUND

Network Calculus [27] is a mature theory proposed fordeterministic performance analysis. It is used to constructarrival and service curve models for the investigated flowsand network nodes. The arrival and service curves are defined

by means of the min-plus convolution.Network Calculus functions mainly belong to non-

decreasing functions and null before 0: F↑ = { f : R+ →R|x1 < x2 ⇒ f (x1) < f (x2),x < 0 ⇒ f (x) = 0}. Two basicoperators on F↑ are the convolution ⊗,

( f ⊗g)(t) = inf0≤s≤t

{ f (t− s)+g(s)}, (1)

and deconvolution �,( f �g)(t) = sup

s≥0{ f (t + s)−g(s)}, (2)

where inf means infimum and sup means supremum.An arrival curve α(t) is a model constraining the arrival

process R(t) of a flow, in which R(t) represents the inputcumulative function counting the total data bits of the flowthat has arrived in the network node up to time t. We say thatR(t) is constrained by α(t) if

R(t)≤ inf0≤s≤t

{R(s)+α(t− s)}= (R⊗α)(t). (3)

A service curve β (t) models the processing capability of theavailable resource. Assume that R∗(t) is the departure process,which is the output cumulative function that counts the totaldata bits of the flow departure from the network node up totime t. There are several definitions for service curve. We saythat the network node offers the min-plus minimum servicecurve β (t) for the flow

R∗(t)≥ inf0≤s≤t

{R(s)+β (t− s)}= (R⊗β )(t), (4)

and offers the strict service curve β (t) iffR∗(t +∆t)−R∗(t)≥ β (∆t), (5)

during any backlog period [t, t +∆t). In addition, in order toevaluate service curves we will use the non-decreasing non-negative closure defined by

[ f (t)]+↑ = max0≤s≤t

{ f (s),0}.

If a flow R(t) of arrival curve α(t) crosses a server with theservice curve β (t), then the output flow R∗(t) can be boundedby the arrival curve α ′(t),

α′(t) = α�β (t) = sup

s≥0{α(t + s)−β (s)} , (6)

It can also be taken as the arrival curve of input flow for thenext node.

Let us assume that the flow constrained by the arrivalcurve α(t) traverses the network node offering the servicecurve β (t). Then, the latency experienced by the flow in thenetwork node is bounded by the maximum horizontal deviationbetween the graphs of two curves α(t) and β (t),

h(α,β ) = sups≥0{inf{τ ≥ 0 | α(s)≤ β (s+ τ)}} . (7)

With these definitions, the worst-case end-to-end delay of theflow is the sum of latency bounds in each network nodes alongits routing.

V. WORST-CASE ANALYSIS FOR AVB TRAFFIC

A. Service Curve for AVB traffic with non-preemption modeIn this section, we will extend the service curve in [23] to

multiple AVB Class Mi (i ∈ [1,NhCBS]) under non-preemption

mode without considering any shaping curves.

4

Fig. 5. Guard bands before TT windows

The service curve for AVB traffic depends on the remainingservice from TT traffic which has the highest priority and isscheduled within specific TT windows according to GCLs. Weassume that the GCL in an output port h has a set of periodicTT windows. Then, the GCL for the output port h is repeatedafter the GCL period ph

GCL, which is the Least CommonMultiple (LCM) of periods of all periodic TT windows inthe output port h, see the example in Fig. 5. Thus, for a givenGCL in an output port, we known the finite number (Nh

T T )of TT windows in the GCL period ph

GCL. It is assumed thatthe time duration of the ith (i ∈ [0,Nh− 1]) TT window inthe output port h is Lh

T T,i, and the relative offset between thestarting time of the ith and jth ( j ∈ [i + 1, i + Nh − 1]) TTwindows within the same GCL period is oh

j,i if taking theith TT window as the reference. Note that oh

j,i equals to 0 ifj = i. Moreover, with the consideration of the non-preemptionmode, in the worst-case, the time duration of the guard bandLh

GB,i before the ith (i ∈ [0,Nh−1]) TT window equals to theminimum value of the maximum transmission time of AVBframes competing in the output port h and the idle time intervalbetween two consecutive TT windows, i.e., (i−1+Nh)%Nhthand ith windows. By merging the guard band effect into theconstruction of the TT aggregate arrival curve as discussed in[23], we have already known the following Lemma,

Lemma 1: The aggregate arrival curve for TT traffic andguard bands with the non-preemption mode in an output porth is given byα

hGB+T T (t) = max

0≤i≤Nh−1{αh

GB+T T,i(t)}

= max0≤i≤Nh−1

i+Nh−1

∑j=i

(LhT T, j +Lh

GB, j) ·C ·⌈ t−oh

j,i +LhGB, j−Lh

GB,i

phGCL

⌉.

(8)Moreover, the service curve for AVB traffic is also related

to the CBS that supplied for each AVB Class Mi. Any timeinterval ∆t can be decomposed by

∆t = ∆t++∆t−+∆t0, (9)where ∆t+ = ∑i ∆t+i (resp. ∆t− = ∑ j ∆t−j ) represents the accu-mulated length of all period where the credit increases (resp.descreases), and ∆t0 = ∑k ∆t0

k is the frozen time of credit Midue to the guard band and TT windows. For example in Fig. 6,there are two rising phases (∆t+1 and ∆t+2 ), two descent phases(∆t−1 and ∆t−2 ) and one frozen phase (∆t0

1 ) of credit M1. Theservice could only be supplied for AVB traffic during thedescent time ∆t− of credit. Then, we extend the service curvefor multiple classes of AVB traffic by the following theorem.

Theorem 1: The min-plus minimum service curve for AVBClass Mi (i∈ [1,Nh

CBS]) with non-preemption mode in an output

Fig. 6. Decomposed interval with non-preemption integration mode

port h is given by

βh[npr]Mi

(t) = idSlMi

[t−

αhGB+T T (t)

C−

cmaxMi

idSlMi

]+↑, (10)

where [npr] represents the non-preemption integration mode,αh

GB+T T (t) is given by Lemma 1, and cmaxMi

is the upper boundof credit Mi given by (12), which proof has to be extended toan arbitrary number of AVB classes and is one of challengesin this paper and will be discussed in Sect. V-B. The proof issimilar as the one given to Theorem 1 in [23], hence will notbe further discussed in this paper.

B. Bounding the credit for AVB trafficIn this section, we bound the credit for AVB traffic. Let

us recall from Sect. III how AVB is transmitted. In TSN, thetransmission of AVB traffic is not only related to the gatestates, but also to CBS. Although TT transmission in non-preemption mode delays AVB traffic, the credits are frozenduring these periods. Therefore, we can say that AVB creditswill not be affected by TT traffic. In fact, the credit value isrelated to the transmission and backlog of AVB frames duringthe time when the respective AVB gates are open.

Theorem 2: (Lower bound of Class Mi) Let lmaxMi

be themaximal frame size of any flow crossing the AVB queue QMi .Then, the credit cMi(t) of Class Mi is lower bounded by [12],

cminMi

=lmaxMi

CsdSlMi ≤ cMi(t). (11)

Proof : Since only when sending a frame of Class Mi thecredit is decreased, the check of the lower bound must be doneat the end of the emission of a frame of Class Mi. Consideringthe evolution of the credit, the minimal value of the creditis reached only when the size of the transmitted frame isthe maximal one. In this situation the transmission period isdefined as ∆t−max

Mi= lmax

Mi/C. Therefore, the credit at the end

of the transmission is cMi(t +∆t−maxMi

) = 0+ sdSlMi · lmaxMi

/C.

Note that the constraint on idSlMi and sdSlMi for any AVBtraffic class satisfies sdSlMi = idSlMi −C [8].

Theorem 3: (Higher bound of Class Mi) Let lmaxBE be the

maximal frame size of a BE flow. The credit cMi(t) of ClassMi is upper bounded by,

cMi(t)≤lmax>i

CidSlMi+(

−lmax>i

C

i−1

∑j=1

idSlM j +i−1

∑j=1

cminM j

)idSlMi

∑i−1j=1 idSlM j −C

= cmaxMi

,

(12)

5

where lmax>i = max j∈[i+1,Nh

CBS]{lmax

M j, lmax

BE } and cminM j

is the lowerbound of credit of Class M j from Theorem 2.

Proof : Consider a time point t and the AVB queue QMi .Let’s note Q≤i

AV B =⋃

1≤ j≤i{QM j} the same or higher priorityAVB queues and Q>i

AV B =⋃

i< j≤NhCBS{QM j} the lower priority

AVB queues.If no frame is being sent at t, it means that for all j ≤ i,

cM j(t)≤ 0, otherwise a frame would be sent.Now consider that a frame is being sent from Q>i

AV B orfrom the BE queue at t. Set s be the start of the e-mission of this frame. Notice that t − s ≤ lmax

>i /C, wherelmax>i = max j∈[i+1,Nh

CBS]{lmax

M j, lmax

BE }. For all j ≤ i, cM j(t) ≤ 0,

otherwise, at time s, an AVB frame from Q≤iAV B would have

been selected for emission. Then we can deduce that for allj ≤ i, cM j(t)≤ idSlM j · lmax

>i /C.Last consider that a frame is being sent from Q≤i

AV B at t.Set x the last time before t where a frame from Q≤i

AV B wasnot being sent. Let’s denote ∆tMi the duration of emissions offrames from queue QMi between x and t. Then it is possibleto upper bound the variation of credit of Class Mi between xand t,

cMi(t)− cMi(x)≤∆tMi · sdSlMi +(t− x−∆tMi) · idSlMi

=−∆tMi ·C+(t− x) · idSlMi .(13)

Let c<i(t) = ∑i−1j=1 cM j(t) denote the sum of credits of AVB

traffic with the priority higher than Mi. Then at any instantbetween x and t either a frame from Class Mi is being sent, inthis case c<i(t) increase at most at speed ∑

i−1j=1 idSlM j . Either

a frame from class with higher priority than Mi is being sent,in this case c<i(t) decrease at least at speed ∑

i−1j=1 idSlM j −C

(all the classes from Q<iAV B gain credit expect one which loses

credit). Then it is possible to upper bound the variation ofc<i(t) between x and t,

c<i(t)−c<i(x)

≤∆tMi ·i−1

∑j=1

idSlM j +(t− x−∆tMi) ·( i−1

∑j=1

idSlM j −C)

=∆tMi ·C+(t− x) ·( i−1

∑j=1

idSlM j −C).

Since ∑i−1j=1 idSlM j −C < 0, then we have

t− x≤ c<i(t)− c<i(x)−∆tMi ·C∑

i−1j=1 idSlM j −C

. (14)

Then from (14), (13) is modified to obtain,

cMi(t)− cMi(x)≤ (c<i(t)− c<i(x))idSlMi

∑i−1j=1 idSlM j −C

−∆tMiC

·(

∑ij=1 idSlM j −C

∑i−1j=1 idSlM j −C

)≤ (c<i(t)− c<i(x))

idSlMi

∑i−1j=1 idSlM j −C

.

We have already proved that for all j ≤ i, cM j(x)≤ idSlM j ·lmax>i /C, therefore c<i(x) ≤ ∑

i−1j=1 idSlM j · lmax

>i /C. Moreover ifcmin

M jis the lower bound of credit of Class M j, then c<i(t) ≥

∑i−1j=1 cmin

M jso to conclude,

cMi(t)≤lmax>i

CidSlMi+(−

lmax>i

C

i−1

∑j=1

idSlM j +i−1

∑j=1

cminM j

)idSlMi

∑i−1j=1 idSlM j −C

.

C. Tighter latency bounds by introducing shaping curves forarrival of AVB traffic

According to Network Calculus, the upper bound latencyof a Class Mi flow τMi[k] in the output port h is given by themaximum horizontal deviation between the aggregate arrivalcurve αh

Mi(t) of intersecting flows of AVB Class Mi and the

service curve βh[npr]Mi

(t) for AVB Class Mi in h,

DhMi[k]

= h(αhMi(t),β h[npr]

Mi(t)), (15)

where the service curve βh[npr]Mi

(t) is from Theorem 1.In this paper, we provide a solution that leads to tighter

aggregate arrival curves for AVB traffic in intermediate nodes,reducing thus the pessimism of WCDs for AVB flows byconsidering the shaping caused by the physical link speed andthe CBS.

The individual arrival curve for the flow τMi[k] in the sourceES (h0) can be given by

αh0Mi[k]

(t) = lMi[k]+lMi[k]

pMi[k]· t (16)

where lMi[k] is the burst of the flow in h0, lMi[k]/pMi[k] is thelong-term rate of τMi[k] sending from the source ES.

The individual arrival curve αhMi[k]

for the flow τMi[k] in theoutput port h of intermediate nodes is calculated from theprevious node port h′ along the path of τMi[k]. Moreover, flowsτMi[k] of Class Mi waiting in the queue of h and from the sameprevious node port h′ are taken as a group, as they cannotsimultaneously arrive because of sharing a physical link, andwill be limited by the CBS. We assume that αh

Mi,h′(t) represents

the grouping arrival curve from the same node port h′ and isgiven by,

αhMi,h′(t) =( ∑

τMi[k]∈[h′,h]

αh′Mi[k]

(t)�δh′Di(t))

∧ (σh′Mi(t)+ lh,max

Mi,h′)∧ (σlink(t)+ lh,max

Mi,h′),

(17)

where αh′Mi[k]

(t)�δ h′DMi

(t) is the output arrival curve consideringthe queuing delay, δD(t) is the burst-delay function whichequals to 0 if t ≤ D and ∞ otherwise, x ∧ y = min{x,y},σlink(t) = C · t is the shaping curve of physical link, lh,max

Mi,h′is

the maximum frame size of flows of Class Mi from h′ to h, andσh′

Mi(t) is the shaping curve of CBS, which will be discussed

in Sect. V-D with the consideration of TT effects.Then the aggregate arrival curve αh

Mi(t) for the output port h

is the sum of all grouping arrival curves of Class Mi. Moreover,it is assumed of no limitation for arriving of flows in the sourceES (h0). Then they can simultaneously arrive and thus theaggregate arrival curve α

h0Mi(t) for the port h0 is the sum of all

individual arrival curves of Class Mi flows.By disseminating the computation of latency bounds along

6

the routing of τMi[k], its WCD is obtained by the sum of delaysfrom its source ES to its destination ES,

DMi[k] = ∑h∈drMi[k]

DhMi[k]

+(h−1) ·dtech, (18)

where dtech is the constant technical latency in a SW.

D. CBS Shaping Curve for AVB trafficShaping curve is a concept which characterizes the maxi-

mum number of bits that are served during a period of time∆t. A server offers a shaping curve σ(t) iff σ(t) could be anarrival curve for all output cumulative function R∗(t), i.e.,

R∗(t +∆t)−R∗(t)≤ σ(∆t), (19)where R∗(t) is the output cumulative function.

Lemma 2: With the non-preemption mode, the strict servicecurve for TT traffic in an output port h is given by,

βhT T (t) = min0≤i≤Nh−1

{ i+Nh−1

∑j=i

βhT DMA(t + t0,LT T, j)

}, (20)

whereβ

hT DMA(t,LT T )

=C ·max{⌊

tph

GCL

⌋Lh

T T , t−⌈

tph

GCL

⌉(ph

GCL−LhT T )

},

and

t0 = phGCL−Lh

T Tj−oh

0,i−ohj,i.

Proof : As discussed in Sect. V-A, for a given GCL in anoutput port, there are Nh number of TT windows in the GCLperiod ph

GCL. Then the service for TT traffic can be taken as Nh

sets of periodic windows with the known length and relativeoffsets.

If considering one set of periodic (phGCL) TT windows of

length LT T , its service in the port h is similar to the TDMAcommunication [28]. In the worst-case time interval ph

GCL−Lh

T T is not supplied for the TT traffic, and has exclusive serviceof the bandwidth for TT during the interval Lh

T T afterwards.Therefore, the service cannot be guaranteed during any timeinterval 0 ≤ ∆t < ph

GCL− LhT T , but can be guaranteed of C ·

(∆t − phGCL− Lh

T T ) in any time interval phGCL− Lh

T T ≤ ∆t <ph

GCL. Then the service curve supplied for the set of periodicTT windows during ∆t can be given by,β

hT DMA(∆t,LT T )

=C ·max{⌊

∆tph

GCL

⌋Lh

T T ,∆t−⌈

∆tph

GCL

⌉(ph

GCL−LhT T )

}.

(21)

Then the service for all Nh sets of periodic TT windows isderived from (21) as well as the relative offsets between twoTT windows from different sets. Taking the ith (i∈ [0,Nh−1])TT window as the reference, we have that the first served frameis from the ith TT window during the backlogged period ∆t.Assume that oh

0,i is the maximum idle time interval before theopening time of the ith window, then the service guaranteedfor the ith set of TT windows is the curve in (21) shifted tothe left with the positive value ph

GCL−LhT T,i−oh

0,i,

βhT T,i,i(∆t) = β

hT DMA(∆t + ph

GCL−LhT T,i−oh

0,i,LT T,i).

Moreover, for the other set of the jth ( j ∈ [i+ 1, i+Nh− 1])traffic windows, with the known of the relative offset oh

j,i byconsidering the ith TT window as the benchmark, the serviceguaranteed for TT traffic in the jth set of windows can begiven by shifting the curve in (21) to the left with the positivevalue ph

GCL−LhT T, j−oh

0,i−ohj,i,

βhT T, j,i(∆t) = β

hT DMA(∆t + ph

GCL−LhT T, j−oh

0,i−ohj,i,LT T, j).

Note that β hT T, j,i(∆t) equals to β h

T T,i,i(∆t) if j = i.Thus if the ith TT window as benchmark, the strict service

for TT traffic is as follows after considering all Nh sets ofperiodic TT windows,

βhT T,i(∆t) =

i+Nh−1

∑j=i

βhT T, j,i(∆t).

Then the strict service curve for TT traffic in an output porth is the lower envelope of β h

T T,i(∆t) by considering each TTwindows in the GCL period as benchmark,

βhT T (∆t) = min0≤i≤Nh−1{β

hT T,i(∆t)}.

Theorem 4: The CBS shaping curve of Class Mi for thenon-preemption mode is given by,

σhMi(t) =

[t− β h

T T (t)C

]+↑· idSlMi + cmax

Mi− cmin

Mi, (22)

where β hT T (t) is given by the Lemma 2, and cmax

Miand cmin

Miare

respectively the upper and lower bounds of credit of Mi givenby (12) and (11).

Proof : For an arbitrary period of time ∆t, the variation ofcredit during ∆t satisfies

∆cMi = cMi(t +∆t)− cMi(t) = ∆t+ · idSlMi +∆t− · sdSlMi

= (∆t−∆t0) · idSlMi −∆t− · (idSlMi − sdSlMi).(23)

In the best-case, the frozen duration is only related to TTwindows, and has nothing to do with guard bands, for the non-preemption integration mode. Considering the strict servicecurve of TT traffic expressed in the Lemma 2, ∆t0 is limitedby,

Rh∗T T (t +∆t)−Rh∗

T T (t) = ∆t0 ·C ≥ βhT T (∆t). (24)

Moreover, ∆t− can be expressed by,Rh∗

Mi(t +∆t)−Rh∗

Mi(t) = ∆t− ·C, (25)

and ∆cMi satisfies the relationship as follows,∆cMi ≥ cmin

Mi− cmax

Mi. (26)

Due to non-decreasing function Rh∗Mi(t) and using the expres-

sions (24), (25), (26) and sdSlMi = idSlMi−C, (23) is modifiedto obtain,

Rh∗Mi(t +∆t)−Rh∗

Mi(t)≤

[∆t− β h

T T (∆t)C

]+↑· idSlMi + cmax

Mi− cmin

Mi.

Considering the definition of the shaping curve in (19), theCBS shaping curve of AVB traffic of Class Mi is given by,

σhMi(t) =

[t−β

hT T (t)

/C]+↑ · idSlMi + cmax

Mi− cmin

Mi.

7

VI. EXPERIMENTAL RESULTS

We have evaluated our proposed improving NetworkCalculus-based WCD analysis for multiple AVB classes inTSN (called PL/CBS-NC/TSN) as follows. PL/CBS-NC/TSNis implemented in C++ using the Java kernel of the RTCtoolbox [29], running on a computer with Intel Core i7-3520MCPU at 2.90 GHz and 4 GB of RAM.

A. Synthetic Test CasesIn this section, our evaluation focuses on a tree topology of

6 ESes and 2 SWs, connected via physical links with rates of100Mb/s, as shown in Fig. 7.

Firstly, we are interested to show the improvement of AVBWCDs in TSN network obtained with our proposed methodby considering physical link and CBS shaping curves incomparison to existing work [23]. The test case (TC1) has5 TT flows and 10 AVB flows of only two classes (M1 andM2), as in order to compare with the model proposed in [23].The physical link rate is 100Mb/s, the average load of linksis about 14.2% and the maximum link load is about 33.1%.The idle slopes of Class M1 and Class M2 are respectively setto 40% and 20% (2).

The WCDs for each AVB flow of TC1 are calculated underdifferent timing analysis models. The results of the comparisonbetween [23] (NC/TSN) and the proposed method (PL/CBS-NC/TSN) are presented in Table I. As we can see, our methodby introducing physical link shaping and CBS shaping curvesis able to significantly reduce the WCDs compared to [23],with 17.0% on average and 26.4% in maximum. Moreover,we also present the results obtained with our method, wherewe have only used physical link shaping but we have not usedCBS shaping curves, denoted with PL-NC/TSN. As we cansee from the third column in Table I, PL-NC/TSN is ableto reduce on average the WCDs obtained with [23] by 5.6%and as much as 9.7% in some cases. Additionally, we givesimulation results in the last column. The simulation resultsdenoted with Sim are useful for validating our approach, as ourobtained WCDs should all be larger than the maximum delaysobtained by simulation, which is the case in our experiments.However, as rare events can be missed, a simulation approachwill not be able to determine the WCDs, hence it is not usefulfor safety critical applications that require safety guarantees.

In the second experiment, we would like to evaluate thescalability of our analysis PL/CBS-NC/TSN with the numberof TT flows and to show the correctness of our analysis bycomparing with simulation results. We use another test case

(2)More detailed information, including GCL for TT flows routesfor each flow etc. can be downloaded from https://zenodo.org/record/2439362#.XBsMS84zbDc.

Fig. 7. The topology of synthetic small test cases

TABLE ICOMPARED RESULTS WITH AND WITHOUT SHAPING CURVE

Flow NC/TSN [23](µs)

PL-NC/TSN(µs)

PL/CBS-NC/TSN(µs)

Sim(µs)

AVB1 5378 5218 4470 2454AVB2 4692 4640 4349 2594AVB3 7128 6775 4851 2812AVB4 3557 3224 2594 1401AVB5 7083 6730 4806 2016AVB6 3780 3392 2651 1508AVB7 7211 6858 4934 2550AVB8 2505 2357 2019 1097AVB9 3557 3224 2594 1398AVB10 3780 3392 2651 1401AVB11 3178 2790 2049 1067AVB12 7211 6858 4934 2248

Fig. 8. WCDs for an increasing number of TT flows: from 5 to 50; AVBflows have been reordered based on their WCDs within each traffic class

(TC2) by extending TC1 such that it has a total of 50 TT flows.The obtained results are visually shown in Fig. 8, where onthe x-axis we have the AVB flows, from AVB1 to AVB12,and on the y-axis we have the WCD values. As shown inFig. 8, the obtained results are grouped by the AVB Classeswith vertical dotted lines and respectively sorted in increasingorder by results from TC1. TC1 is depicted with a red triangleand TC2 with a blue cross. As expected, the WCDs of the AVBflows grow with the increasing number of TT flows shown inFig. 8. In addition, the average WCD (horizontal dotted lines)of Class M1 is smaller than M2, since the AVB class of higherpriority has larger bandwidth guarantee, i.e., idSlM1 > idSlM2 .

Fig. 9. Network topology of the Orion CEV

8

Fig. 10. WCDs decrease with increasing idle slopes

B. Evaluation on a Larger Realistic Test CaseFor the final experiment, we use a larger real-world test

case (TC3), adapted from the Orion Crew Exploration Vehicle(TC3) [30] by using the same topologies and TT flows, andconsidering rate-constrained (RC) flows as AVB flows. Weinvestigate the scalability of our method and the influence ofvaried idle slopes on multiple classes of AVB traffic. CEV hasa topology consisting of 31 ESes, 15 SWs, 188 dataflow routes,as shown in Fig. 9, connected by dataflow links transmittingat 1 Gbps, and running 100 TT flows, 25 AVB flows of ClassM1, 25 AVB flows of Class M2, 24 AVB flows of Class M3and 13 AVB flows of Class M4 (including multicast flows).

The expectation is that higher values for the idle slopewill allow the AVB flows to pass faster through switches,reducing their WCDs. For AVB traffic, the idle slope idSlMi iscommonly established for granting just enough credit requiredto preserve the accumulated rate of all AVB Class Mi flows.In this experiment, the reference of idle slopes of M1, M2, M3and M4 are respectively 40%, 20%, 10% and 5%. Then weincrease the idSlMi by keeping idSlM j ( j 6= i) of other AVBclass unchanged. The WCDs obtained by considering varyingidle slopes for each AVB traffic Class are shown respectivelyin Fig. 10(a), (b), (c) and (d). Moreover, the results are sortedin increasing order and the average WCD under each case ofidle slope is given by the horizontal dotted lines. As expected,the latency for AVB traffic declines with the increasing idleslopes, since the larger the idle slope, the higher the bandwidthguarantee is. For example, idSlM1 increasing from 40% to 45%leads to a decrease of WCD of AVB Class M1 traffic with 5.6%on average.

VII. CONCLUSION

The TSN IEEE task group is defining new extensions ofEthernet, devoted to real-time and safety-critical applicationareas. These types of applications require a method to boundthe worst-case delays of a given configuration. This paperhas presented an improved Network Calculus-based approachby considering the limitations from physical link rate and

the output of CBS, which are modeled as shaping curves tocompute tighter bounds. Moreover, the timing analysis methodhas be extended to arbitrary number of AVB traffic classes inthe case of a TSN/GCL+CBS architecture, i.e., a system whereone or more queues are scheduled in a Time-Triggered way,with exclusive access to the output link (excluding gating),and the others are shaped by a CBS credit (like in AVB).

Our analysis is, to the best of our knowledge, the first oneto handle with any number of AVB classes. We have evaluatedthe proposed approach on both synthetic and realistic testcases. The experimental results and the comparison to theexisting approaches show that the Network Calculus approachis a viable approach for the analysis of TSN. Our approachprovides safe upper bounds on WCDs, reduces the pessimismof the analysis (tighter WCD bounds), and is scalable to handlelarge problem sizes.

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